Preconditioning the Pressure Tracking in Fluid Dynamics by Shape Hessian Information

Abstract

Potential flow pressure matching is a classical inverse design aerodynamic problem. The resulting loss of regularity during the optimization poses challenges for shape optimization with normal perturbation of the surface mesh nodes. Smoothness is not enforced by the parameterization but by a proper choice of the scalar product based on the shape Hessian, which is derived in local coordinates for starshaped domains. Significant parts of the Hessian are identified and combined with an aerodynamic panel solver. The resulting shape Hessian preconditioner is shown to lead to superior convergence properties of the resulting optimization method. Additionally, preconditioning gives the potential for level independent convergence.

Computing quantities of interest for random domains with second order shape sensitivity analysis

“…Due to their much higher complexity, shape Hessians are not so often applied and the literature about computations with second order shape derivatives is restricted (see, e.g., [11,12,14,23]). Nevertheless, the shape Hessian underlies the study of stability issues in shape optimization (see [1,3,4,6,7,8,13,21,25,28,31] for examples in imaging, tomography, fluid mechanics, aircraft construction, etc.). For the sake of readability, we present two academic but representative examples for which the expressions of the shape derivatives remains simple.…”

Efficient shape optimization for certain and uncertain aerodynamic design

“…Rather than more general investigations on aerodynamic shape optimization [28], we assume that a finite parametrization of the shape to be optimized is given. The vector y is the state vector (velocities, pressure,.…”

Airfoil design for compressible inviscid flow based on shape calculus

“…For the viscous case of incompressible Stokes and Navier-Stokes flows such a symbol derivation can be found in all detail in Schmidt (2010), Schmidt and Schulz (2009). For potential flow, a similar analysis can be found in Eppler et al (2009) and for the inviscid compressible case see Arian and Ta'asan (1996), Arian and Vatsa (1998). Thus, only a brief overview of this method is given here.…”